an optimality criterion for supersaturated designs with quantitative factors

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Journal of Statistical Planning and Inference

Journal of Statistical Planning and Inference 142 (2012) 1780–1788

0378-37

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jspi

An optimality criterion for supersaturated designs withquantitative factors

Chao Huang a,c, Dennis K.J. Lin b, Min-Qian Liu c,n

a School of Mathematics, University of Manchester, P.O. Box 88, Manchester M13 9PL, UKb Department of Statistics, The Pennsylvania State University, University Park, PA 16802, USAc Department of Statistics, School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

a r t i c l e i n f o

Article history:

Received 28 September 2011

Accepted 1 February 2012Available online 7 February 2012

Keywords:

Geometrical isomorphism

Supersaturated design

g wordlength pattern

58/$ - see front matter & 2012 Elsevier B.V. A

016/j.jspi.2012.02.003

esponding author.

ail address: [email protected] (M.-Q. Liu).

a b s t r a c t

A supersaturated design (SSD) is a factorial design in which the degrees of freedom for

all its main effects exceed the total number of distinct factorial level-combinations

(runs) of the design. Designs with quantitative factors, in which level permutation

within one or more factors could result in different geometrical structures, are very

different from designs with nominal ones which have been treated as traditional

designs. In this paper, a new criterion is proposed for SSDs with quantitative factors.

Comparison and analysis for this new criterion are made. It is shown that the proposed

criterion has a high efficiency in discriminating geometrically nonisomorphic designs

and an advantage in computation.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

Factorial designs are commonly used in scientific studies. In such studies, a number of fixed levels (settings) areselected for each factor, and some level-combinations are then chosen in an experiment. The factors can be nominal(qualitative) or quantitative. A nominal factor is a factor whose levels can be ordered free; whereas, a quantitative factorrequires its levels to be in order. Thus, different types of data analysis methods are required. The purpose of analyzing anexperiment with nominal factors is to investigate whether there exist differences in treatment means and if they do, whichtreatment means are different. Many methods, such as ANOVA or multiple comparison testing, are often used fortreatment comparison. On the other hand, methods of response surface exploration are often used to deal with the data ofexperiments with quantitative factors. It is thus obvious that we need different criteria for design classification and designselection for different types of factors.

For designs with nominal factors, the design properties should be invariant to level permutation within factors. This isreferred to as combinatorial isomorphism. Designs with nominal factors have received a great deal of attention in theliterature, e.g. the maximum resolution (Box and Hunter, 1961), minimum aberration (Fries and Hunter, 1980), clear effects(Wu and Chen, 1992), and general minimum lower order confounding (Zhang et al., 2008) criteria for regular designs, andthe minimum G2-aberration (Tang and Deng, 1999), generalized minimum aberration (GMA, Deng and Tang, 1999; Ma andFang, 2001; Xu and Wu, 2001), minimum moment aberration (Xu, 2003), discrete discrepancy (Fang et al., 2003), minimumprojection uniformity (Hickernell and Liu, 2002), and minimum hybrid aberration (Pang and Liu, 2010) for nonregulardesigns. Note that all these criteria are invariant to level permutation and only applicable to designs with nominal factors.

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dx.doi.org/10.1016/j.jspi.2012.02.003

mailto:[email protected]

dx.doi.org/10.1016/j.jspi.2012.02.003

Table 1Two combinatorially isomorphic designs each with six runs and three three-level factors.

Design I Design II

A B C A B C

0 0 0 1 0 0

0 1 1 1 1 1

1 0 2 f0;1,2g2f1;0,2g 0 0 2

1 2 0 on factor A 0 2 0

2 1 2 2 1 2

2 2 1 2 2 1

A

B

C

A

B

C

Fig. 1. Geometric structures for the two designs in Table 1. (a) Design I. (b) Design II.

C. Huang et al. / Journal of Statistical Planning and Inference 142 (2012) 1780–1788 1781

For designs with quantitative factors, in which their level permutations could result in different data analysis, the abovecriteria may not be appropriate. Table 1 shows two designs with six runs and three three-level factors, where each columnrepresents a factor and each row represents an experimental run. These designs are combinatorially isomorphic since oneis obtained by applying the permutation f0;1,2g-f1;0,2g on factor A of the other. However, if these levels are quantitative,their geometric structures are apparently different as shown in Fig. 1. For example, there is a ‘‘central point’’ in Fig. 1(a) forDesign I whilst no ‘‘central point’’ in Fig. 1(b) for Design II.

Supersaturated design (SSD) is a kind of factorial design in which the number of runs is insufficient to estimate all themain effects. It has recently received much interest mainly because of its potential in factor screening, namely, to efficientlyidentify a few active factors among many candidate factors at the initial step; see for example, Lin (1993, 1995, 2003). Multi-level and mixed-level SSDs arise, when two-level SSDs could not meet some specific experimental demands, see e.g. Liu andLin (2009) for a motivating example and Sun et al. (2011) for some most recent construction results. There are many criteriaspecially defined for evaluating multi-level and mixed-level SSDs, such as, the Eðd2

Þ (Lu and Sun, 2001), w2ðDÞ (Yamada andMatsui, 2002) and Eðf

NODÞ (Fang et al., 2003) criteria, see Section 2.3 below for the relevant definitions. Of course, these

criteria are also invariant to level permutation within the factors and only applicable to SSDs with nominal factors.Take the two designs in Table 1 as an example. It can be seen that they take the same values of Eðd2

Þ, EðfNODÞ and w2ðDÞ,

respectively, and cannot be discriminated under any of these criteria though they have the different geometric structuresas shown in Fig. 1. It is thus desirable to have a new criterion to evaluate multi-level and mixed-level SSDs withquantitative factors.

In this paper, we propose a new criterion, specifically for SSDs with quantitative factors, to distinguish thegeometrically nonisomorphic designs. Section 2 introduces some terminology and existing aberration criteria. The newcriterion is proposed in Section 3. Section 4 provides some analysis and comparison results. Section 5 presents someconcluding remarks.

2. The aberration criteria

In this section, we first review relevant notations and terminology, then discuss the b WLP and GMA criterion due toCheng and Ye (2004), followed by the generalized WLP and GMA criterion due to Xu and Wu (2001). Existing criteria forSSDs and their relations are also reviewed here. A new criterion will be proposed in next section.

C. Huang et al. / Journal of Statistical Planning and Inference 142 (2012) 1780–17881782

2.1. Notations and terminology

Let D be a design space of the full factorial design with n factors and v design points, denoted as Dðv,s1s2 � � � snÞ, wherev¼ s1s2 � � � sn, the levels of the ith factor are set at Gi ¼ f0;1, . . . ,si�1g,i¼ 1, . . . ,n. An n-factor design D with N runs is said tobe in D if for any run x 2 D, x 2 D, and we denote this design as DðN,s1s2 � � � snÞ. A design DðN,s1s2 � � � snÞ is called balanced ifall the levels occur equally often in each factor, and called supersaturated when

Pni ¼ 1ðsi�1Þ4N�1. The following two

definitions are due to Cheng and Ye (2004).

Definition 1. Let D be a design in the design space D. The indicator function FDðxÞ of D is a function defined on D whichcalculates the number of appearances of point x in design D for x 2 D.

Definition 2. Let D1 and D2 be two designs from the same design space D. They are said to be geometrically isomorphic ifone can be obtained from the other by relabeling the factors and/or reversing the level order of one or more factors.

It is clear that the indicator function FDðxÞ uniquely represents a design. Let T ¼ G1 � � � � � Gn. For the ith factor of ann-factor design, define a set of orthogonal contrasts Pi

0ðxÞ,Pi1ðxÞ, . . . ,P

isi�1ðxÞ such that

Xx2f0;1,...,si�1g

PiuðxÞP

ivðxÞ ¼

0 if uav,

si if u¼ v:

(

Then an orthonormal contrast basis (OCB) on D is defined as

PtðxÞ ¼Yn

i ¼ 1

PitiðxiÞ

for t¼ ðt1,t2, . . . ,tnÞ 2 T and x¼ ðx1,x2, . . . ,xnÞ 2 D. It is obvious that

Xx2D

PtðxÞPuðxÞ ¼0 if tau,

v if t¼ u,

(

where t and u are the elements in the set T . If Pi0ðxÞ ¼ 1 for all i, then we call fPtðxÞg a statistical orthonormal contrast basis

(SOCB). When PijðxÞ is a polynomial of degree j for j¼ 0;1, . . . ,si�1 and i¼ 1;2, . . . ,n, the SOCB is called an orthogonal

polynomial basis (OPB, see Draper and Smith, 1998, Chapter 22).

2.2. b wordlength pattern

For the indicator function FDðxÞ and OCB fPtðxÞ,t 2 T g, Cheng and Ye (2004) showed that

FDðxÞ ¼Xt2T

btPtðxÞ

for all x 2 D, and the coefficients fbt,t 2 T g are uniquely determined as

bt ¼1

v

Xx2D

PtðxÞ: ð1Þ

Particularly, b0 ¼N=v with 0¼ ð0;0, . . . ,0Þ. Then the b WLP and aberration criterion for factorial designs with quantitativefactors can be defined as follows.

Definition 3. Let D be a DðN,s1s2 . . . snÞ with quantitative factors in design space D and let fPtðxÞg be an OPB. The b WLPðb1ðDÞ, . . . ,bK ðDÞÞ is defined as

biðDÞ ¼X

JtJ1 ¼ i

ðbt=b0Þ2: ð2Þ

The GMA criterion is to sequentially minimize bi for i¼ 1;2, . . . ,K , where K ¼Pn

i ¼ 1ðsi�1Þ and JtJ1 ¼Pn

j ¼ 1 tj.

Remark 1. Note that in this definition, if we let fPtðxÞg be an SOCB, and replace biðDÞ by

aiðDÞ ¼X

JtJ0 ¼ i

ðbt=b0Þ2 for i¼ 1;2, . . . ,n, ð3Þ

where JtJ0 ¼ i represents the number of nonzero elements in t, then we have the generalized WLP ða1ðDÞ, . . . ,anðDÞÞ and theGMA criterion due to Xu and Wu (2001) for designs with nominal factors. And this aberration criterion is invariant to levelpermutation as well as the choice of contrasts.

The only distinction between these two WLPs are the norms of t used in (2) and (3), which reflect the difference inordering effects for quantitative and nominal factors. For a given design, the sum of its ai’s is the same as the sum of its bi’s.Cheng and Ye (2004) showed that this sum is a constant for designs that have the same run sizes and replication patterns.Furthermore, Cheng and Ye (2004, Corollary 4.1) proved that two designs with different b WLPs must be geometrically


nonisomorphic. And for most cases, two geometrically nonisomorphic designs also have distinct b WLPs, see e.g. Tsai et al.(2006). Thus, the b WLP can be used to classify and select designs with quantitative factors.

2.3. Existing criteria for SSDs and their relations

SSDs are nonregular factorial designs, in which, orthogonality is not obtainable. Most existing criteria for SSDs measurethe non-orthogonality combinatorially between two factors. We next review some existing criteria for multi-level andmixed-level SSDs.

For a design DðN,s1 � � � snÞ, define

w2ði,jÞ ¼Xu,v

ðnðijÞuv�N=ðsisjÞÞ2=ðN=ðsisjÞÞ,

where nðijÞuv is the number of runs in which the ith and jth factors take the level-combination uv. Then the w2ðDÞ criterionproposed by Yamada and Matsui (2002) is to minimize

w2ðDÞ ¼X

1r io jrn

w2ði,jÞ:

And the EðfNODÞ criterion proposed by Fang et al. (2003) is to minimize

EðfNODÞ ¼

X1r io jrn

ðw2ði,jÞN=ðsisjÞÞ

,n

2

� �:

When s1 ¼ � � � ¼ sn ¼ s, EðfNODÞ reduces to the Eðd2

Þ proposed by Lu and Sun (2001).Note that Xu and Wu (2005) used the a2ðDÞ as a criterion for multi-level SSDs. Recently, Liu et al. (2006) generalized the

w2ðDÞ criterion to the so-called minimum w2 criterion, and provided statistical justification for both a2ðDÞ and w2ðDÞ. Theyshowed that

a2ðDÞ ¼ w2ðDÞ=N, ð4Þ

when D is a balanced design, and furthermore

a2ðDÞ ¼ w2ðDÞ=N¼ s2 n

2

� �Eðf NODÞ

.N2¼ s2 n

2

� �Eðd2Þ

.N2, ð5Þ

when all factors have s levels. Obviously, all these criteria for SSDs attempt to minimize the aberration between twofactors, and they are invariant to level permutation. Hence they are not applicable to designs with quantitative factors.

3. A new aberration criterion for SSDs

Since SSDs are typically used in factor screening experiments, i.e., to identify few active factors among many potentialfactors, the first- and second-order aberrations are dominated. We thus propose the g WLP below to characterize such animportant feature for quantitative factors.

Definition 4. Let D be a DðN,s1s2 � � � snÞ with quantitative factors in design space D and let fPtðxÞg be an OPB. Then the newWLP ðg1ðDÞ, . . . ,gK 0 ðDÞÞ, called the g WLP, is defined as

giðDÞ ¼X

JtJ0 r2,JtJ1 ¼ i

ðbt=b0Þ2, i¼ 1;2, . . . ,K 0, ð6Þ

where K 0 ¼max fsiþsj�2 : iaj, i,j¼ 1, . . . ,ng, and bt is defined in (1). The new GMA criterion is to sequentially minimize gi

for i¼ 1;2, . . . ,K 0.

Example 1. For SSDs with quantitative factors, Definition 4 only considers the main effects and two-factor interactions.Consider the case of a three-level factorial design with quantitative factors. Each factor has three orthogonal polynomialcontrasts: Pi

0,Pi1,Pi

2, i¼ 1, . . . ,n, where Pi0 represents a constant term, denoted as ‘‘0’’; Pi

1 is the linear contrast, denoted as‘‘l’’; and Pi

2 is the quadratic one, denoted as ‘‘q’’. A main effect or two-factor interaction can be denoted by two contrasts.Then the order of effect importance according to the g WLP is

0l¼ ¼ l0b0q¼ ¼ q0¼ ¼ llb lq¼ ¼ qlbqq,

where b means ‘‘more important than’’ while ¼ ¼ means ‘‘as important as’’; and 0l and l0 indicate linear main effects,0q and q0 indicate quadratic main effects, lq linear-by-quadratic interaction, etc. An important feature of SSD is itsbalance property. For a balanced design, it can be shown that bt ¼ 0 for all t with only one nonzero element: For thethree-level case, this implies that the linear main effects 0l and l0, and the quadratic main effects 0q and q0, havezero correlations with the constant term. So g1ðDÞ ¼ 0, and thus g2ðDÞ can be easily computed. The g WLP now reduces to


ð0,g2ðDÞ,g3ðDÞ,g4ðDÞÞ, and the order of effect importance needed to be considered is simplified to

llb lq¼ ¼ qlbqq:

From (2) and (6), it is obvious that

giðDÞ ¼ biðDÞ for i¼ 1;2

for any DðN,s1s2 � � � snÞ design D with quantitative factors in design space D. Note thatPK 0

i ¼ 1 giðDÞ ¼ a1ðDÞþa2ðDÞ anda1ðDÞ ¼ 0 for a balanced design. We thus have the following theorems regarding the properties of the g WLP.

Theorem 1. Let D be a balanced design in design space D, then the sum of giðDÞ’s in the g WLP equals the a2ðDÞ defined

in (3), i.e.

XK 0i ¼ 1

giðDÞ ¼ a2ðDÞ:

Theorem 2. Let D1 and D2 be two geometrically isomorphic designs in design space D. Then their g WLPs are identical.

Theorem 1 shows that balanced designs with the same a2ðDÞ value take the same sum of gðDÞ’s. For any given designwith quantitative factors, level permutation may generate designs with different geometric structures, but with the samea2ðDÞ value (see Remark 1), and hence these resulting designs have the unique sum of giðDÞ’s. While Theorem 2 shows thattwo designs with different g WLPs must be geometrically nonisomorphic, thus one can differentiate geometricallynonisomorphic designs by comparing their g WLPs. The proof of Theorem 2 can be straightforwardly obtained, by applyingTheorem 3.1 of Cheng and Ye (2004).

Remark 2. For conventional factorial designs (where the number of experimental runs exceeds the number of parametersin the model), the b WLP can be used for quantitative factors to differentiate their geometrical isomorphism. For SSDs,however, all existing criteria are only good for designs with nominal factors. The proposed g WLP appears to be the firstcriterion to consider both—it is useful for SSDs with quantitative factors. Note that the b WLP is lengthy and much morecomplicated to evaluate than the g WLP. This can be seen by, when n42, K ¼

Pni ¼ 1ðsi�1Þ4K 0 ¼max fsiþsj�2 :

iaj, i,j¼ 1, . . . ,ng, where K and K 0 are the total number of elements of b and g WLP, respectively. Moreover, b WLPinvolved with many high-order interactions which are normally irrelevant in SSDs.

The next section contains some comparison and analysis results for applying the g and b WLPs to discriminategeometrically nonisomorphic SSDs.

4. Comparisons and analysis

The basic strategy for applying the g WLP is as follows:

Step 1.
Select a good SSD according to a traditional criterion. As we can see from (4) and (5), an SSD having a smaller valueof w2ðDÞ or Eðf
NODÞ also has a smaller value of a2ðDÞ, which implies that the design has a smaller sum of giðDÞ’s. Thus

w2ðDÞ or EðfNODÞ optimal SSDs are preferred in this stage.

Step 2.
For each SSD selected in the first stage, apply level permutations to each factor, and then compute the g WLPs ofthe resulting designs. Based on Theorem 2, the resulting designs with different g WLPs are geometricallynonisomorphic.
Step 3.
From the resulting designs, find the optimal ones under the new GMA criterion.
In the following, three examples of multi-level SSDs (Examples 2–4) and one example of mixed-level SSD (Example 5)will be demonstrated for selecting and comparing the geometrically nonisomorphic designs. The b WLPs will also becomputed for comparison.

Example 2. Consider the design in Table 4 of Fang et al. (2004) as an illustration. It is an EðfNODÞ- and w2ðDÞ-optimal Dð6,35

Þ

design. The design is given in Table 2, where the five factors are headed by A, B, C, D and E, respectively.Note that there are a total of six permutations among three levels. The six permutations can be divided into three pairs

as shown in Table 3. In each pair of the three kinds of permutations, level 0 and level 2 exchange their positions while level1 stays at its position, i.e., 0;1,2-2;1,0, 1;2,0-1;0,2 and 2;0,1-0;2,1. Therefore, within each pair, one permutation isthe reverse of the other, hence, only one is needed in generating geometrically nonisomorphic designs. These three pairs ofpermutations are denoted as a, b and c in the first row of Table 3. For this design, permutations are applied to each factor tosearch for all geometrically nonisomorphic designs. Then following the steps given above, we have the results presented inTable 4. Note that in Table 4, ‘‘baaaa’’ represents the design obtained by applying permutations b,a,a,a and a (as illustratedin Table 3), respectively to the five factors A, B, C, D and E, and so on. It can be seen from Table 4 that both the g and b WLPsagree upon the identical optimal designs.

Table 3Six permutations of three levels.

a b c

0 1 2

1 2 0

2 0 1

y y y

2 1 0

1 0 2

0 2 1

Table 4

Example 2—optimal designs according to g and b WLPs for the design in Table 2.

Subject g WLP b WLP

Optimal WLP (0, 0.625, 3.75, 0.625) (0, 0.625, 7.5, 8.8281, 4.6875, 10.625, 4.6875, 1.0156, 0, 1.5313)

Optimal designs baaaa, caaaa, bbbca, cbbca, bcbca, baaaa, caaaa, bbbca, cbbca, bcbca,

ccbca, bbcca, cbcca, bccca, cccca ccbca, bbcca, cbcca, bccca, cccca

Table 2

Dð6,35Þ from Fang et al. (2004).

A B C D E

0 0 0 0 0

0 1 1 1 1

1 0 2 2 1

1 2 0 1 2

2 1 2 0 2

2 2 1 2 0

Table 5

Example 3—Dð8,44Þ from Xu and Wu (2005).

A B C D

0 0 0 0

1 1 1 1

2 2 2 2

3 3 3 3

0 1 2 3

1 2 3 0

2 3 0 1

3 0 1 2


Note that, since g1ðDÞ ¼ b1ðDÞ ¼ 0, we only need to compute three elements g2ðDÞ, g3ðDÞ and g4ðDÞ for each g WLP, buthave to compute nine elements for each b WLP. Moreover, the results show that the g WLP performs efficiently in findingthe optimal SSDs.

Example 3. Next we consider a four-level Dð8,44Þ design, given in Table 5. This design is optimal under the GMA criterion

defined in Remark 1, hence it is also optimal under the EðfNODÞ and w2ðDÞ criteria.

For the four-level case, there are a total of 24 permutations which can be divided into 12 different pairs as shown inTable 6. The numbers a,b, . . . ,l in the first row of the table represent these 12 pairs. As shown in Table 6, with levelexchanges 0-3, 1-2, 2-1 and 3-0, we could get one permutation from the other in each pair. Take the twopermutations in column f as an example, with the level exchanges, 0, 3, 2, 1 becomes 3, 0, 1, 2, accordingly. Similar toExample 2, Table 7 is obtained.

From Table 7, we see that both the g and b WLPs find the identical optimal designs. However, the g WLP can save muchmore computational efforts than the b WLP.

There are often more than one optimal multi-level SSDs under the EðfNODÞ or w2ðDÞ criteria, these designs have the same

a2ðDÞ value (cf. Eq. (5)). Then if all factors are quantitative, they will have the same sum of giðDÞ’s. Thus we can further

Table 6Example 3—12 pairs of permutations among four levels.

a b c d e f g h i j k l

0 0 0 0 0 0 1 1 1 1 1 1

1 1 2 2 3 3 0 0 2 2 3 3

2 3 1 3 1 2 2 3 0 3 0 2

3 2 3 1 2 1 3 2 3 0 2 0

y y y y y y y y y y y y

3 3 3 3 3 3 2 2 2 2 2 2

2 2 1 1 0 0 3 3 1 1 0 0

1 0 2 0 2 1 1 0 3 0 3 1

0 1 0 2 1 2 0 1 0 3 1 3

Table 7


Subject g WLP b WLP

Optimal WLP (0, 0.04, 0, 5.92, 0, 0.04) (0, 0.04, 0, 9.36, 0, 11.12, 0, 8.52, 0, 1.96, 0, 0)

Optimal designs dlgb, lgbd, bdlg, gbdl dlgb, lgbd, bdlg, gbdl

Table 8

Example 4—Dð9,34 sÞ ð1rsr7Þ.

1 2 3 4 5 6 7

0 2 1 0 2 0 0 0 1 1 2 0 1 0 1 2 0 1 0 2 1 2 0 1 0 0 2 1

0 0 2 1 0 2 1 0 2 0 0 0 1 1 2 0 1 0 1 2 0 1 0 2 1 2 0 1

1 2 0 1 0 0 2 1 0 2 1 0 2 0 0 0 1 1 2 0 1 0 1 2 0 1 0 2

0 1 0 2 1 2 0 1 0 0 2 1 0 2 1 0 2 0 0 0 1 1 2 0 1 0 1 2

1 0 1 2 0 1 0 2 1 2 0 1 0 0 2 1 0 2 1 0 2 0 0 0 1 1 2 0

1 1 2 0 1 0 1 2 0 1 0 2 1 2 0 1 0 0 2 1 0 2 1 0 2 0 0 0

2 0 0 0 1 1 2 0 1 0 1 2 0 1 0 2 1 2 0 1 0 0 2 1 0 2 1 0

2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Table 9Example 4—optimal designs according to g WLP based on the designs in Table 8.

g WLP Optimal design

1 2 (aabb baba) 2 3 (aabb baba) 3 4 (aabb aabb)

4 5 (aabb baba) 5 6 (aabb aabb) 6 7 (aabb baba)

(0, 1.25, 5.5, 1.25) 1 4 (baab baba) 2 5 (baab baba) 3 6 (abaa abcb)

4 7 (baab baba) 1 6 (aaab abcb) 2 7 (aaab abcb)

1 7 (baba aaba)

1 3 (baaa bbbb) 2 4 (baab bbcb) 3 5 (baab bbcb)

(0, 1.5, 5, 1.5) 4 6 (baab bbcb) 5 7 (baab bbcb) 1 5 (abba abab)

2 6 (abbb abab) 3 7 (abba abab)


discriminate their resulting designs from level permutations, and select the optimal ones under the new GMA criterion, asdemonstrated in the next example.

Example 4. A class of EðfNODÞ-optimal Dð9,34s

Þ designs for s¼ 1, . . . ,7 obtained in Fang et al. (2004) are listed in Table 8.There are seven designs in the table, each design is an OAð9,34

Þ, and every two designs form an EðfNODÞ-optimal Dð9,38

Þ

design. There are altogether ð72Þ such Dð9,38Þ designs. For each design, permutations given in Table 3 are applied to each

factor, and the optimal design under the new GMA criterion is shown in Table 9. Note that in this table, ‘‘1 2 (aabb baba)’’represents the SSD obtained by applying permutations a,a,b,b,b,a,b and a to the respective eight factors of the first twodesigns in Table 8. From Table 9, we can see that there are only two different kinds of optimal designs among the ð72Þdesigns, the better ones are those with the g WLP of (0, 1.25, 5.5, 1.25), we can select any of these, e.g. 1 2 (aabb baba), as anoptimal Dð9,38

Þ design with quantitative factors for further study.

Table 10

Example 5—Dð6,35210Þ from Yamada et al. (2006).

A B C D E F G H I J K L M N O

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 2 2 0 0 0 0 1 1 1 1 1 1

2 1 2 2 0 0 1 1 1 0 0 0 1 1 1

0 1 1 1 1 1 0 1 1 0 1 1 0 0 1

1 2 2 0 1 1 1 0 1 1 0 1 0 1 0

2 2 0 1 2 1 1 1 0 1 1 0 1 0 0

Table 11


Subject g WLP b WLP

Optimal WLP (0, 40.625, 23.750, 0.625) ð0;40,440;1940, . . .Þ

Optimal designs

(total of 31)

baaaa, caaaa, bbbbb,

bbbbb, . . .

baaaa, caaaa, bbbbb,

bbbbb, . . .


Example 5. Consider a mixed-level w2ðDÞ-optimal SSD Dð6,35210Þ obtained in Yamada et al. (2006). This design has 6 runs

and 15 factors, 5 with 3 levels and 10 with 2 levels. For two-level factors, there are only two permutations, 0, 1 and 1, 0,which are geometrically identical to each other, so the geometrical nonisomorphism only depends on the permutations ofthe three-level factors. The design is listed in Table 10 and the selected optimal designs are listed in Table 11. In Table 11,the optimal designs are represented by the five permutations on the five three-level factors. For example, ‘‘baaaa’’represents the design obtained by applying permutations b,a,a,a and a to the five three-level factors, A, B, C, D and E,accordingly. Totally 31 optimal geometrically nonisomorphic designs were selected under the g and b WLPs, respectively,while only four of them are listed in Table 11 for saving space. Thus, in this mixed-level SSD case, the g and b WLPs agreeon the same optimal geometrically nonisomorphic designs again.

5. Concluding remarks

SSD is a special kind of factorial design, all existing criteria for assessing SSDs are invariant to level permutation withinthe factors and thus only applicable to the case of nominal factors. Design criteria for quantitative factors are not as well-developed as those for nominal ones, however.

In this paper, we propose the g WLP for SSDs with quantitative factors. The properties of this new WLP are explored, and itis applied to discriminating geometrically nonisomorphic SSDs and finding the optimal ones. Several examples have beenprovided for illustration, some comparisons are also made with the b WLP. It is shown that the g WLP is able to efficientlyidentify the optimal SSDs. Although the b WLP can also classify geometrically nonisomorphic SSDs and find the optimal ones,the computation of the b WLP is much more complicated. Specifically, given an SSD DðN,snÞ, the computation intensity of theg WLP is C2

ns2 since this WLP considers any 2 of n factors and each factor has ‘‘s’’ orthogonal polynomial contrasts, while thecomputation intensity of the b WLP is sn since it needs to consider all the factors with ‘‘s’’ orthogonal polynomial contrasts. Asfor an SSD, the number of factors n is large, the computation could be dramatically reduced by using the g WLP.

The g WLP can be further extended in the following way. For two interaction contrasts Pt1ðxÞ and Pt2

ðxÞ withJt1J0 ¼ Jt2J0 ¼ 2 and Jt1J1 ¼ Jt2J1, Pt1

ðxÞ is considered to be more important than Pt2ðxÞ if 9tj1

�ti1 9o9tj2�ti2 9, where

t1 ¼ ð0, . . . ,0,ti1 ,0, . . . ,0,tj1,0, . . . ,0Þ and t2 ¼ ð0, . . . ,0,ti2 ,0, . . . ,0,tj2

,0, . . . ,0Þ. Namely, for interaction effects with the same order,those of two medium-order effects are more important than those of one high-order and one low-order effects. Take thequantitative four-level factorial design as an example, we have the original order of effect importance according to the g WLP:

0l¼ ¼ l0b ll¼ ¼ 0q¼ ¼ q0b lq¼ ¼ ql¼ ¼ 0c¼ ¼ c0bqq¼ ¼ lc¼ ¼ clbqc¼ ¼ cqbcc,

where ‘‘c’’ denotes the cubic polynomial contrast. While the improved order of effect importance changes to

0l¼ ¼ l0b ll¼ ¼ 0q¼ ¼ q0b lq¼ ¼ ql¼ ¼ 0c¼ ¼ c0bqqb lc¼ clbqc¼ ¼ cqbcc,

which has one more order than the original one. Because this extension only increases the number of different orders, not ofthe bt’s, so it also holds the advantage of computation convenience.

Acknowledgements

This work was supported by the Program for New Century Excellent Talents in University (NCET-07-0454) of China andthe National Natural Science Foundation of China Grant 10971107.


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an optimality criterion for supersaturated designs with quantitative factors

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